SOME MARKOV-SWITCHING MODELS FOR THE TORONTO STOCK EXCHANGE - MUNICH PERSONAL REPEC ARCHIVE
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Munich Personal RePEc Archive Some Markov-Switching Models for the Toronto Stock Exchange Accolley, Delali Le Mans Université 19 March 2021 Online at https://mpra.ub.uni-muenchen.de/108072/ MPRA Paper No. 108072, posted 07 Jun 2021 10:19 UTC
Some Markov-Switching Models for the Toronto Stock
Exchange ∗
Delali Accolley
accolleyd@aim.com
Laboratoire de recherche en Économie GAINS, Le Mans Université, Le Mans, France
March 19, 2021
Abstract
This research motivates the use of Markov chains in modeling financial time series.
Then, it explains the returns and the volatility on the Toronto Stock Exchange (TSX)
using some Markov-switching models. These models are: the conditional capital asset
pricing model, the conditional Sharpe model, and the exponential autoregressive model
with state-dependent heteroscedasticity. It also tests for cointégration between the TSX
and some other major exchanges, relying on the first-order and the second-order Markov
chains.
The asymmetry, the multiple peaks, or the fat tails in the distribution of the returns
on the TSX and on the other exchanges indicates they could not be modelled as random
realizations from a single normal distribution. The switching regressions turn out to have
a greater explanatory power and provide further understanding of the TSX.
Keywords: Econometrics, Finance, Markov Chain.
JEL: G0, 016
Résumé
Cette recherché justifie l’utilisation des chaînes de Markov dans la modélisation des
séries chronologiques financières. Ensuite, elle explique les rendements et la volatilité sur
la Bourse de Toronto (TSX) en utilisant quelques modèles de Markov à changement de
régime. Ces modèles sont : le modèle conditionnel d’évaluation des actifs financiers, le
modèle conditionnel de Sharpe et le modèle autorégressif exponentiel avec une hétérosce-
dasdacité conditionnelle qui dépend du régime. Elle teste également la cointégration entre
le TSX et d’autres bourses, en s’appuyant sur les chaînes de Markov de premier et de
second ordre.
L’asymétrie, les nombreux pics ou l’épaisseur des queues de la distribution des rende-
ments sur la TSX et sur les autres bourses indiquent qu’ils ne peuvent pas être modélisés
comme étant des réalisations aléatoires provenant d’une seule distribution normale. Il
s’avère que les régressions avec changement de régime ont un plus grand pouvoir explicatif
et fournissent une meilleure compréhension du TSX.
Mots clés : Econométrie, Finance, chaîne de Markov.
JEL : G0, 016
∗
I am grateful to Messrs Jill Scullion and John Andrew from the TSX for having provided me with
some information on the global industry classification standard.CONTENTS i
Contents
Some Abbreviations and Acronyms ii
Non-Technical Summary / Sommaire Non-Technique iii
1 Introduction 1
2 Motivation 3
3 The Markov-Switching Model 6
3.1 First-Order Markov Chain . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 Higher-Order Markov Chain . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.3 Dependent Mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4 The Conditional CAPM 9
4.1 The Method of Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.2 The Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2.1 The First-Order Markov-Switching CAPM . . . . . . . . . . . . . 13
4.2.2 The Second-Order Markov-Switching CAPM . . . . . . . . . . . 19
5 The Accounting of the Market Return 22
5.1 The Method of Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.2 The Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.2.1 The First-Order Markov-Switching Sharpe Model . . . . . . . . . 25
5.2.2 The Second-Order Markov-Switching Sharpe Model . . . . . . . . 27
6 The Exponential Autoregressive Model 28
6.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6.2 The Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
7 The Cointegration of International Stock Markets 32
7.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
7.2 The Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
7.2.1 The First-Order Markov-Switching Cointegrating Relations . . . 35
7.2.2 The Second-Order Markov-Switching Cointegrating Relations . . 37
8 Conclusion 40
Appendices 41
A The Data 41ii CONTENTS
B Some Basic Concepts 43
B.1 Kernel Density Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 43
B.2 The Linear Regression Model . . . . . . . . . . . . . . . . . . . . . . . . 45
B.2.1 The Significance Tests . . . . . . . . . . . . . . . . . . . . . . . . 46
B.2.2 The Model Selection Criteria . . . . . . . . . . . . . . . . . . . . 47
B.2.3 An Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
B.3 Stationarity and Cointegration . . . . . . . . . . . . . . . . . . . . . . . 49
B.3.1 Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
B.3.2 Cointegration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
B.4 The Derivation of the CAPM . . . . . . . . . . . . . . . . . . . . . . . . 54
B.4.1 The Optimization Problem . . . . . . . . . . . . . . . . . . . . . 54
B.4.2 The Consumption versus the Traditional CAPM . . . . . . . . . 57
B.4.3 The Market Excess Return and the Risk Aversion . . . . . . . . . 58
Some Abbreviations and Acronyms
ADF Augmented Dickey-Fuller
AEG Augmented Engle-Granger
AIC Akaike Information Criterion
AR Auto-Regressive
ARCH Auto-Regressive Conditional eteroskedasticity
ARMA Auto-Regressive Moving Average
ASX Australian Securities Exchange
BIC Bayesian Information Criterion
Bovespa Bolsa de Valores do Estado de São Paulo
BSE Bombay Stock Exchange
CAC Cotation Assistée en Continu
CDF Cumulative Distribution Function
DAX Deutscher Aktienindex
DCC Dynamic Conditional Correlation
EM Expectation-maximization
FTSE Financial Times Stock Exchange
GARCH Generalized Auto-Regressive Conditional eteroskedasticity
HMM Hidden Markov Model
IBEX Índice Bursátil Español
ISEQ Irish Stock Exchange Quotient
NASDAQ National Association of Securities Dealers Automated Quotations
NYSE New York Stock Exchange
OLS Ordinary Least Squares
PDF Probability Density Function
SENSEX Sensitive Index
SMI Swiss Market Index
S&P Standard & Poor’s
TSX Toronto Stock Exchange
UK United Kingdom
US United StatesCONTENTS iii
Non-Technical Summary
Motivation Financial time series are not normally distributed as often assumed. For
example, on Toronto Stock Exchange (TSX), the likelihood of extreme negative or ex-
treme positive returns is higher than in a normal distribution. Besides, whereas the
graph of a normal distribution is bell-shaped, the graph of the actual distribution of the
returns on the TSX is not symmetrical and has several peaks.
Objectives This research seeks to explain returns across the TSX by accommodating
the asymmetry in their distributions and the higher likelihood of both extreme negative
or extreme positive values. It also purports to study appropriately the existence of a
long-run equilibrium relationship between stock prices on the TSX and stock prices on
some other major exchanges.
Methodology I have assumed the returns are random realizations from a mixture of
normal distributions, each distribution having its own mean and variance. Each of these
means and variances are associated to one of the recurring states of a stock market (the
bull or the bear markets).
Key Contributions I have modeled returns, volatility, and the state variable simul-
taneously across the sectors of the TSX. I have taken into account the recurring states
of stock markets, while investigating the existence of long-run equilibrium relationship
between the TSX and some other major exchanges.
Findings Materials (particularly, gold) and utilities are the defensive sectors of the
TSX. During periods of low volatility, the financial, the energy, and the materials sectors
account for more than half of the market returns. However, during periods of high
volatility, the share of the energy sector drops considerably. Volatility on the TSX has
also turned out to be asymmetric.
In bull markets which are periods of low volatility, the autocorrelation of returns
could be high or low depending on whether the returns are low or extremely high. On
the other hand, in bear markets which are periods of high volatility, the autocorrelation
is low because returns are often negative and extremely low.
Modeling and identifying the recurring states of stock markets have revealed the
existence of long-run equilibrium relationships between the stock prices on the TSX
and the stock prices on some other major exchanges that include the New York Stock
Exchange, the Tokyo Stock Exchange, and the London Stock Exchange. Whatever the
method of estimation used, I have not found any evidence of equilibrium relationship
between the stock prices on the TSX and the stock prices on either the NASDAQ or the
Bolsa de Madrid.
It also turned out that dissociating the trends of stock markets from the turning
points could improve considerably the explanatory power of the models.iv CONTENTS
Sommaire Non-Technique
Motivation Contrairement à ce qu’on suppose généralement, les séries chronologiques
financières ne sont pas des variables qui suivent une distribution normale. Par exemple,
sur la Bourse de Toronto (TSX), la probabilité d’observer des rendements négatifs ou
positifs extrêmes est plus élevée que celle d’une distribution normale. Par ailleurs, alors
que la représentation graphique d’une distribution normale a la forme d’une cloche, celle
de la distribution des rendements du TSX n’est pas symétrique et a plusieurs pics.
Objectifs Cette recherché veut expliquer les rendements sur le TSX en tenant compte
de leur distribution asymétrique et de la probabilité plus élevée d’observer des ren-
dements positifs ou négatifs extrêmes. Elle vise également à étudier convenablement
l’existence d’une relation d’équilibre de long terme entre le cours des actions sur le TSX
et le cours des actions sur d’autres bourses.
Methodologie J’ai supposé que les rendements sont des réalisations aléatoires tirées
d’un mélange de distributions normales ayant chacune sa propre moyenne et variance.
Chacune de ces moyennes et variances est associée à un des états récurrents d’un marché
boursier (marché haussier ou marché baissier).
Contributions majeures J’ai modélisé les rendements, la volatilité et la variable
d’état simultanément à travers les secteurs du TSX. J’ai pris en compte les états récur-
rents des marchés boursiers en étudiant l’existence de relations d’équilibre de long terme
entre le TSX et les autres bourses.
Résultats Les matériaux (particulièrement l’or) et les services publics sont les secteurs
défensifs du TSX. Durant les périodes de faible volatilité, les secteurs de la finance,
de l’énergie et des matériaux génèrent plus de ela moitié des rendements du marché.
Toutefois, quand la volatilité est forte, la contribution du secteur de l’énergie baisse
considérablement. La volatilité sur le TSX s’avère être asymétrique.
Dans les marches haussiers qui sont des périodes de faible volatilité, l’autocorrélation
des rendements peut être élevée ou faible selon que les rendements sont faibles ou ex-
trêmement élevés. Par contre, dans les marchés baissiers qui sont des périodes de haute
volatilité, l’autocorrélation est faible parce que les rendements y sont souvent négatifs
et extrêmement bas.
Le fait de modéliser et d’identifier les états récurrents des marches boursiers a révélé
l’existence d’une relation d’équilibre de long terme entre les cours boursiers sur le TS
et les cours des actions sur d’autres bourses, dont la Bourse de New York, la Bourse de
Tokyo et la Bourse de Londres. Quelle que soit la méthode d’estimation utilisée, je n’ai
trouvé aucune preuve de relation d’équilibre entre les cours des actions sur le TSX et les
cours des actions sur le NASDAQ ou la Bourse de Madrid.
Il s’est avéré que le fait de dissocier les tendances des marchés boursiers de leurs
tournants peut améliorer considérablement le pouvoir explicatif des modèles.1
1 Introduction
Toronto Stock Exchange (TSX) is the largest stock market in Canada and also one of
the ten most important in the world. Companies listed on the TSX can be classified into
eleven sectors according to their main activities. These sectors are: consumer discre-
tionary (e.g., automobiles, consumer durables, hotels), consumer staples (e.g., beverage,
food, tobacco), energy (e.g., oil and gas production, refining, or storage, coal), financial
(e.g., banks, capital markets, insurance), health care (e.g., biotechnology, health care
services, pharmaceuticals), industrial (e.g., consulting services, security services, trans-
portation), information technology (e.g., electronic components or equipment, technol-
ogy distributors), material (e.g., aluminum, copper, gold), real estate (e.g., equity real
estate investment trusts, real estate development), telecommunication service (e.g., ad-
vertising, broadcasting, entertainment), and utilities (e.g., electricity, gas, water). For
further details on this classification, see MSCI Barra and Standard & Poors (2018).
The indicators used to track the overall performance of the TSX and the performance
of the sectors of this exchange are the various Standard & Poor’s (S&P)/TSX indices
and sub-indices. These indices are weighted averages of the trading prices of selected
stocks. Their growth rates give the returns on the market and on a sector portfolio.
Two main trends characterize a stock exchange: the bear and the bull markets. A
bear market is a less frequent period of financial turbulence where returns are generally
low and highly volatile. On the other hand, a bull market is a period of widespread
optimism and euphoria among investors. Returns across most sector are generally high
and less volatile, during a bull market (Vendrame, Guermat, and Tucker, 2018). Each
of these two trends, in its own way, interacts with the economic activity. For example,
an incipient bear market that has started during a period of economic expansion, leads
to a recession, which in turn causes a grizzly bear market. Hamilton and Lin (1996) find
that the stock market volatility leads the economic activity by one month. They also
find that, in turn, recession is the single and major cause of stock volatility. 1
As Hamilton and Susmel (1994), Abdymomunov and Morley (2011), and Vendrame,
Guermat, and Tucker (2018) point out, ignoring the state prevailing in the market leads
to wrong estimates and forecasts when fitting models to financial time series. As a
matter of fact, the parameters of models explaining returns and their volatility are not
constant but instead are time-varying due to the alternation of various states on financial
markets.
Therefore, this research aims at modeling returns on the TSX and their volatility
conditionally on the state prevailing on this market. A way of doing this is through the
use of a Markov chain. A Markov chain offers the possibility of modeling in a flexible
and general way the probability of moving from an unobserved state to the other. For
this reason, it can be used to explain and predict the alternation or the persistence of
1
Hamilton and Lin (1996) measure the economic activity using the change in the natural logarithm
of the industrial production index of the economy of the United States for the period ranging from
January 1965 to June 1993. The stock market excess returns are the changes in the natural logarithm
of the S&P 500 plus the dividend yields minus the monthly equivalent of the 3-month Treasury bill
yield.2 1 INTRODUCTION
stock market trends. One can therefore induce time variation in the parameters of an
econometric model by allowing its likelihood to depend on a Markov chain. Conditional
models generated in this way are referred to as Markov-switching models.
Earlier research that have used Markov-switching models to explain returns on the
TSX and their volatility include Van Norden and Schaller (1993). They deem that
the existence of speculative bubbles in asset prices could explain their fluctuations.
As a consequence, the states of stock markets (i.e., crashes and booms) could stem
from the apparent deviation of asset prices from their market fundamental price. The
fundamental price of an asset is the sum of the current dividend it pays and all the
expected future dividends discounted by the risk-free rate. Using data on the TSX for
the period 1956-1989, Van Norden and Schaller calculate inter alia the ex ante and
the ex post probability of a market crash. The ex post probability shows spikes that
correspond to actual crashes. They also find that the ex ante probability rises before a
crash, which suggests that deviations from the fundamentals could predict the states of
stock markets.
In this research, I have used Markov chains to estimate: (1) the conditional capital
asset pricing model (CAPM) for the sectors of the TSX, (2) the contribution to the
market return of each of the sectors making up the TSX (the conditional Sharpe model),
(3) the relation between the variance of the returns on the TSX and their autocorrelation
using a conditional exponential autoregressive model, and (4) the conditional bivariate
relationship between the stock prices on the TSX and the stock prices on some other
major stock exchanges.
The rest of this paper is organized as follows. Section 2 motivates the use of the
Markov-switching models when dealing with financial time series. The main reason is
that the histogram or the graph of the nonparametric probability distribution of finan-
cial time series is not symmetrical and bell-shaped as the unconditional normal distri-
bution used to represent them implies. Section 3 briefly introduces to Markov chains
and Markov-switching models. The presentation of these two tools follows Hamilton
(1994) and mostly Zucchini and MacDonald (2009). Sections 5 through 7 present the
econometric models used to explain returns and their volatility as well as the empirical
evidence. Section 8 concludes this research.
The particularity of some of the empirical investigations is the use of both the first-
order and the second order Markov chains as well as the simultaneous estimations of
the models using state-dependent multivariate normal distributions. The second-order
Markov chain adds precision to the models, as it dissociates the two main trends of stock
markets, which are the bear and the bull markets, from their turning points.
In Section 4, I have fitted mixtures of state-dependent multivariate normal models.
Out of their component expected values and variance-covariance matrices, I have esti-
mated the parameters of the conditional CAPM, simultaneously for all the sectors of
the TSX. It emerges from this investigation that the consumer staples and the utilities
sectors and the gold sub-industry offer a hedge against market downturns.
In Section 5, I have estimated the shares of each of the sectors of the TSX in the
market return. It turns out that during periods of low volatility, the financial, the
energy, and the materials sectors account for more than half of the market returns. On3
the other hand, during periods of high volatility, it is the financial sector followed by
the consumer discretionary and the materials that generate most of the market returns.
The importance of the contribution of most sectors depends on the state prevailing in
the market. These investigations have also confirmed the existence of asymmetry in the
volatility on the TSX.
In Section 6, I have investigated the relation between the variance of the daily ;re-
turns and their autocorrelation. The exploratory analysis of the data shows that when
the autocorrelation is high the volatility is low, but the autocorrelation is not always
high when the volatility is low. This observation leads me to model the relation between
the current and the lagged daily returns on the TSX using an exponential autoregressive
process with state-dependent conditional heteroskedasticity. I turns out that during a
bull markets (for the same level of volatility), when the returns are low their autocor-
relation is high and when they are extremely high their autocorrelation is low. During
bear markets where extreme negative returns are frequent, their autocorrelation is low.
In Section 7, I have investigated the existence of cointegation (i.e. a long-run equi-
librium relationship) between the TSX and some other majors exchanges. Unlike some
authors who have studied cointegration assuming structural breaks in the data, I have
assumed that the bivariate relationships between the stock prices on the TSX and the
stock prices on each of the other 15 exchanges in my sample depend instead on the
recurring states of the stock markets. To estimate the parameters of these bivariate
relations, I have fitted a state-dependent multivariate normal distribution to the data.
The New York Stock Exchange, the Tokyo Stock Exchange, the London Stock Exchange,
and some other exchanges have turned out to be cointegrated with the TSX. But, I have
not found any evidence of cointegration between the TSX and either the NASDAQ or
the Bolsa de Madrid.
2 Motivation
When fitting models to financial time series, it is often assumed that they are normally
distributed. A random variable that is normally distributed has a bell-shaped density
curve, which is symmetrical about its mean. A density curve is a graphical representation
of a probability distribution. Furthermore, a normally distributed variable is completely
described by its mean and its variance, which are assumed to be constant.
There are various ways of checking whether a financial time series is actually normally
distributed. One could either compare its kernel density curve to that of a normal
distribution or compute statistics describing the shape of its distribution. A kernel
density curve is the graph of probability values estimated without assuming priorly a
parametric statistical distribution. (I provide a note on kernel density estimation in
Appendix B.1.) Two descriptors of the shape of a distribution are the skewness (S) and
the kurtosis (K). For observations rt (t = 1, . . . , T ) with mean r̄, these descriptors are
expressed as follows:4 2 MOTIVATION
PT
1 (rt − r̄)3
S = T h t=1
2
i3
PT 2 2
t=1 (rt − r̄)
PT
(rt − r̄)4
K = T hP t=1 i2 .
T 2
t=1 (rt − r̄)
Skewness measures the asymmetry of a distribution. A negative skewness indicates
that the tail on the left-hand side (lhs) of a density curve is fatter than the one on its
right-hand side (rhs). On the other hand, a positively skewed distribution has a longer
tail on its rhs. As for the kurtosis, it indicates whether the tails on either side of a
density curve are thinner or fatter than those of a normal distribution. The kurtosis of
a normal distribution is 3. The excess kurtosis is therefore defined as the kurtosis minus
3. So, a distribution with thinner tails has a negative excess kurtosis and a distribution
with fatter tails has a positive excess kurtosis. One can use simultaneously these two
descriptors of the shape of a distribution to test for normality. A way of doing this is
to perform Jarque-Bera test. The joint null hypothesis of this test is S = 0 and K = 3,
and its alternative hypothesis is either S 6= 0 or K 6= 3. The statistic of Jarque-Bera
(JB) test, which follows a χ2 distribution with 2 degrees of freedom, is
" #
T 2 (K − 3)3
JB = S + ∼ χ2 (2).
6 4
Figure 2.1 compares the kernel density curves of returns across the TSX to those of
normal distributions. It appears that assuming a normal distribution for returns across
the TSX is not quite appropriate. The kernel density curves of the returns in sectors
such as financial, industrial, information technology, and telecommunication service are
slender than those of the normal distributions generated using their respective means
and variances. Besides, unlike the density curve of a normal distribution, they are not
bell-shaped and many of them have more than one peak.
Table 2.1 reports the estimates for the skewness, the kurtosis, and the Jarque-Bera
stastitic across the TSX, inter alia. Returns on the TSX are negatively skewed, except
for the gold sub-industry. This means that, with the exception of the gold sub-industry,
the likelihood of extreme negative returns is higher than that of extreme positive returns
across the TSX. The highest skewness is observed in the financial sector. All the returns
across the TSX have a positive kurtosis. This means they have more outliers, i.e.
extreme values, than a normally distributed variable. The returns in the financial and
in the information technology sectors display the highest kurtosis. Finally, all the Jarque-
Bera statistics are greater than their 5% critical value, which equals 5.991. Therefore,
the null hypothesis that the returns are normally distributed cannot be accepted. In
an earlier investigation, Episcopos (1996) conclude, using daily data ranging from July
30, 1990 to June 30, 1994, that returns across the TSX follow distributions that deviate
from the normal distribution.1 Consumer Discretionary 2 Consumer Staples 3 Energy 4 Financial
0.12
0.12
0.06
0.08
0.08
0.08
0.04
Probability
Probability
Probability
Probability
0.04
0.04
0.04
0.02
0.00
0.00
0.00
0.00
−20 −10 0 5 −20 −10 0 5 −20 0 10 20 −30 −10 0 10
Return Return Return Return
5 Gold 6 Industrial 7 Information Technology 8 Materials
0.04
0.06
0.08
0.03
0.04
0.04
Probability
Probability
Probability
Probability
0.02
0.04
0.02
0.02
0.01
0.00
0.00
0.00
0.00
−20 0 20 40 −20 −10 0 10 −40 −20 0 20 −30 −10 10
Return Return Return Return
9 Telecommunication Serv. 10 Utilities 11 60 Largest Companies 12 Market
0.12
0.12
0.08
0.08
0.08
0.08
Probability
Probability
Probability
Probability
0.04
0.04
0.04
0.04
0.00
0.00
0.00
0.00
−20 −10 0 10 −20 −10 0 5 −20 −10 0 5 −20 −10 0 5
Return Return Return Return
5
Kernel Estimates Normal Distribution
Figure 2.1: Comparison of the Kernel Density Curves of Returns across the TSX to those of Normal Distributions.6 3 THE MARKOV-SWITCHING MODEL
Table 2.1: Characteristics of Returns, TSX, 1998:M1-2017:M12.
Mean Standard Skewness Excess Jarque
Sector/Segment Deviation Kurtosis Bera
Consumer Discretionary .488 4.170 -0.644 1.053 28.154
Consumer Staples .884 3.532 -0.371 .760 9.849
Energy .318 6.909 -0.440 1.227 26.112
Financial .629 4.825 -1.266 7.997 5 156.879
Industrial .502 5.388 -0.986 2.730 241.325
Information Technology .267 9.949 -0.577 3.326 379.571
Materials .272 7.421 -0.618 2.458 163.192
Gold .114 10.443 0.386 1.630 49.084
Telecommunication Service .458 5.156 -0.502 1.731 61.642
Utilities .300 3.811 -0.400 1.126 20.595
The 60 Largest Companies .386 4.438 -1.269 4.524 986.335
Market return .370 4.328 -1.364 4.880 1 231.664
The two parameters describing a normal distribution, which are the mean and the
variance, are assumed to be constant. Actually, the average value of stock returns
vary depending on the general market sentiment, which can be bearish (i.e., negative),
bullish (i.e., positive), mixed, or neutral. Besides, an empirical regularity characterizing
financial time series referred to as volatility clustering proves wrong the assumption
that the variances of stock returns are constant parameters. Volatility clustering is the
observation that periods of unusually high volatility in financial time series are followed
by quieter periods, as high (low) negative or positive returns tend to follow high (low)
returns. As a consequence, the variance, a measure of volatility, is not a constant
parameter but a random variable fluctuating around a constant mean.
To take into account volatility clustering, (generalized) autoregressive conditional
heteroskedasticity (ARCH) models popularized by Engle (1982) and Bollerslev (1986)
are used to estimate the variance of financial time series. But, as Hamilton and Sus-
mel (1994) showed using weekly returns on the NYSE, these models impute a lot of
persistence to stock volatility and give relatively poor forecasts. An alternative way of
specifying these models is to consider high and low volatility periods as distinct states
of nature and then assume that returns are random realizations from a state-dependent
mixture of normally distributions, each having its own mean and variance.
3 The Markov-Switching Model
3.1 First-Order Markov Chain
A latent (i.e., an unobserved) state variable St (t = 1, . . . , T ) that assumes only one
of the discrete values 1, . . . , m, is said to be a Markov chain if it satisfies the following
property
Pr (St+1 = j|St = i, . . . , S1 = g) = Pr (St+1 = j|St = i) = γij . (3.1)3.2 Higher-Order Markov Chain 7
According to (3.1), the probability of moving from a state i at time t to a state j at
time t + 1 does not depend on the past realizations of St . The conditional probabilities
γij can be compacted into an m × m matrix Γ referred to as transition probability
matrix. Each row of Γ represents the probability of moving from a given state i to all
the other possible states and consequently sums to unity.
Γ1′ = 1′ , (3.2)
where 1′ is the transpose of an m × 1 vector of ones.
According to (3.2), 1′ and 1 are respectively the a right eigenvector and the corre-
sponding eigenvalue of Γ. Generally speaking, a nonzero column vector v is said to be
a right eigenvector of a square matrix Γ and the scalar λ its eigenvalue, if Γv = λv.
On the other hand, a nonzero row vector u is said to be the left eigenvector of Γ, if
uΓ = λu. A right and a left eigenvectors differ from each other, unless Γ is a symmetric
matrix. It thus follows that the left eigenvector corresponding to the eigenvalue 1 differs
from 1, unless Γ = Γ′ .
The left eigenvector of Γ corresponding the eigenvalue 1 is of particular interest
because it defines the stationary distribution of St . To see that, let ut denote the
unconditional probability distribution of St ,
ut = [Pr(St = 1), . . . , Pr(St = m)]
ut+1 = ut Γ. (3.3)
The unconditional probability distribution ut is stationary if, inter alia, its expecta-
tion E (ut ) equals E (ut+1 ) = u, which implies that (3.3) becomes u = uΓ. (See Ap-
pendix B.3, for a definition of stationarity.) Thus, the left eigenvector of Γ associated
to the eigenvalue 1 corresponds to the stationary distribution of the St . Since u is a
vector of the unconditional probabilities of all the possible values that St can assume,
one expects it to sum to unity. Putting together the condition for stationarity and the
constraint u1′ = 1 gives the folling relation
u (Im − Γ + J) = 1, (3.4)
where Im and J are respectively the identity matrix of size m and an m × m matrix of
ones.
3.2 Higher-Order Markov Chain
When the probability of moving from a state i at time t to a state j at time t + 1
depends also on some earler realizations of St , the latter latent variable is said to follow
a higher-order Markov chain. As an example, a second-order Markov chain satisfies the
following property
Pr (St+1 = j|St = i, . . . , S1 = g) = Pr (St+1 = j|St = i, St−1 = h) = γhij . (3.5)
The latent process described by (3.5) can be transformed into a first-order Markov chain
by constructing a 1×2 vector from a combination of St−1 and St . For a two-state Markov8 3 THE MARKOV-SWITCHING MODEL
chain, there are four possible combinations of the values that St−1 and St can assume,
which are: (1) [St−1 = 1, St = 1], (2) [St−1 = 1, St = 2], (3) [St−1 = 2, St = 1], and
(4) [St−1 = 2, St = 2]. These four combinations are respectively labeled St∗ = 1, . . . , 4.
If the newlyy defined latent variable St∗ is in state 1, which corresponds to the vector
[St−1 = 1, St = 1], the next period, it will be impossible to move to state 3, which
corresponds to the vector [St = 2, St+1 = 1], or to move to state 4 corresponding to
[St = 2, St+1 = 2]. The reason is that the second element of the vector corresponding to
St∗ = 1, which is St = 1, does not match the first element of the vectors associated to
∗
St+1 ∗
= 3 or St+1 = 4, which is St = 2. When St∗ = 1, the only two options available are:
either to remain in this state with probability c1 or to move to state 2 with probability
1 − c1 . The same reasoning applies to states St∗ 2 through 4.
The transition probability matrix of the newly defined first-order Markov chain St∗
resulting from the transformation of the second-order Markov chain St is therefore
c1 1 − c1 0 0
0 0 c2 1 − c2
Γ∗ =
c3 1 − c3 0
.
0
0 0 c4 1 − c4
Hamilton and Lin (1996, p 578) show how to transform a third-order Markov chain
into a first-order one. In this research, I only deal with first-order and second-order
Markov chains.
3.3 Dependent Mixture
A time series Rt is said to be generated by an m-state Markov-switching model if, in
addition to either (3.1) or (3.5), its conditional probability satisfies the following property
Pr (Rt |Rt−1 , . . . , R1 , Xt , . . . , X1 , St , . . . , S1 ; θ)
= Pr (Rt |Rt−1 , . . . , Rt−k , Xt , . . . , Xt−k , St , . . . , St−k ; θ) , (3.6)
≡ Pr (Rt |Zt ; θ)
′
where Zt = Rt−1 , . . . , Rt−k , X′t , . . . , X′t−k , St , . . . , St−k , Xt and θ are respectively vec-
tors of exogenous variables and parameters.
In (3.6), the distribution of Rt depends not only on its own past k realizations and
the k + 1 most recent values assumed by some exogenous explanatory variables but
also on the states of an unobserved Markov process. It is referred as state-dependent
probability distribution because of it depends on the states of a Markov chain.
The state-dependent probability given by (3.6) is a general specification for all the
models I will be dealing with. The vector of parameters θ as well as the transition
probabilities can be estimated either by directly maximizing the likelihood of the ob-
servations or by implementing the expectation-maximization (EM) algorithm (for more
details, see Hamilton, 1990; Zucchini and MacDonald, 2009, among others).9
4 The Conditional CAPM
The CAPM predicts a linear relationship between the expected excess return on a finan-
cial asset and the market excess return. An excess return (also known as risk premium)
is the difference between the returns on a risky and a risk-free assets. In Appendix B.4,
I present a derivation of the CAPM
E(Rit − Rf t ) = βi E(Rmt − Rf t ), (4.1)
where E is the expectation operator. The variables Rf t , Rit , and Rmt respectively denote
the returns on the risk-free asset, on the risky asset i (i = 1, 2 . . . ), and on the market.
The slope parameter βi , which is the sensitivity of the excess return on an asset i to the
excess market return, is referred to as systematic risk or (market) beta.
The CAPM has received little empirical support as an intercept term and other
explanatory variables have turned out to be instrumental in explaining the excess return
on an asset (see Fama and French, 2004, among others). 2
Besides, the slope parameter βi is not constant over time as the CAPM posits (Ja-
gannathan and Wang, 1996; Abdymomunov and Morley, 2011; Vendrame, Guermat,
and Tucker, 2018). For instance, cyclical stocks, as opposed to defensive stocks, tend to
perform well when the market trends upwards and tend to perform poorly when it trends
downwards. As a consequence, the beta of a portfolio of cyclical assets is expected to
be higher when the market trends upwards than when it trends downwards.
There are various ways of modeling conditionally the CAPM. Henriksson and Merton
(1981) propose the following deterministic approach to test for the ability of an investor
to correctly forecast the sign of the excess market return,
Rit − Rf t = αi + β1i (Rmt − Rf t ) + β2i max(Rmt − Rf t , 0) + εit . (4.2)
Model (4.2) distinguishes between two states, which are: (1) the up market where Rmt −
Rf t > 0 and βi = β1i + β2i , and (2) the down market where Rmt − Rf t < 0 and βi = β1i .
Pettengill, Sundaram, and Mathur (1995) and Lam (2001) also use a dummy a
variable to distinguish between up markets and down markets. But, unlike Henriksson
and Merton (1981), Pettengill, Sundaram, and Mathur (1995) use a two-pass approach to
study instead the conditional relationship between the realized returns on risky portfolios
and the market beta
Rit = γot + γ1t δt βi + γ2t (1 − δt )βi + εit , (4.3)
where δt = 1, if Rmt − Rf t > 0, and δ = 0, if Rmt − Rf t < 0. The explanatory variable
in (4.3), which is βi , results from a prior estimation of (4.1) for each individual asset
using time series. The intercept terms γ0t are the returns on the risk-free asset. The
2
Some of these explanatory variables referred to as the CAPM anomalies are: the earnings-price ratio
(i.e., the capital gain and dividend on a stock relative to its market value), the debt-equity ratio (i.e.,
the total liabilities of a company divided by the value of shareholders’ equity), and the book-to-market
ratio (i.e., the ratio of the book value of a company’s common equity to its market value).10 4 THE CONDITIONAL CAPM
parameters γ1t and γ2t are the market excess returns. Consequently, γ1t is expected
to be positive since it is associated to the dummy variable δ = 1 and γ2t is expected
to be negative since it is associated with δ = 0. To check for the conditional CAPM,
Pettengill, Sundaram, and Mathur (1995) propose to perform some one-sided significance
tests on the averages of the T cross-sectional estimates of γ1t and γ2t . If the averages γ1t
and γ2t are respectively significantly positive and negative, this means there is actually
a positive relationship between the returns and the beta of assets during periods of
positive excess market return and a negative relationship during periods of negative
excess market returns.
The problem with using either (4.2) or (4.3) to estimate the CAPM conditionally on
the state prevailing in the market is the deterministic nature of the threshold used. For
example, a one-off positive excess return does not mean that a market is trending upward
or is in a bull state. Abdymomunov and Morley (2011) and Vendrame, Guermat, and
Tucker (2018) defined the bear and the bull states by rather distinguishing between two
states of the volatility in the market.
Abdymomunov and Morley (2011) estimate the CAPM distinguishing between the
low and high volatility states of the market. The switch between these two states follows
a Markov chain. They estimate the conditional CAPM in three main steps. The first
main step of their three-pass approach consists in decoding the states of the market
using the relation
Rmt − Rf t = µm,St + σm,St zmt , (4.4)
where the market innovation, σm,St zmt , follows a zero-mean state-dependent normal
distribution They estimate the standard deviations of this distribution, σm,St , along with
the expected values of market return, µm,St , using the maximum likelihood method. The
second pass consists in estimating the conditional betas using the dynamic conditional
correlation (DCC) model of Engle (2002). To explain the DCC model, let’s consider the
following two variables that folows a normal distribution
Rmt − Rf t µmt σm,m,t σm,i,t zmt
= + , (i = 1, 2, . . . ), (4.5)
Rit − Rf t µit σm,i,t σi,i,t zit
2
where the beta of asset i equals σm,i,t /σm,m,t , as shown in relation (B.26). Knowing the
2 2
two conditional variances, σm,m,t and σi,i,t , and the conditional correlation coefficient,
one can estimate the dynamic covariance σm,i,t using the following relation
Σit = Dit Cit Dit ,
where Σit denotes the conditional variance-covariance matrix defined in relation (4.5),
the matrix Dit consists of the square root of the diagonal elements of Σit , and the
matrix Cit is the matrix of dynamic conditional correlations. To obtain the consecutive
2
values of σm,m,t 2 , Vendrame, Guermat, and Tucker (2018) assume they follows
and σi,i,t
a generalized autoregressive conditional heteroskedasticity (GARCH) process (which is
described in Section 5 and in Section 6). The estimator for the DCC proposed by Engle4.1 The Method of Estimation 11
(2002) is defined as follows.
Qit = (1 − a − b)Ci + aet−1 e′t−1 + bQi,t−1
Cit = (diagQit )1/2 Qit (diagQit )1/2
where Qit and Ci denote respectively the matrices of pseudo and unconditional covari-
ances (correlations), and et−1 is a vector of standardized residuals. The third pass of
the approach of Vendrame, Guermat, and Tucker (2018) consists in estimating the bear
and the bull risk premia using the individual fixed effects panel model. Using data on
25 portfolios of stocks over the sample period 1926-2015 and the sub-sample 1980-2015,
they find (1) the bear risk premium to be negative and significant, and (2) the bull risk
premium to be positive and significant.
Abdymomunov and Morley (2011) also estimate a Markov-switching CAPM. They
assume volatility to be constant within each of the two or three possible states of the
market. But, unlike Vendrame, Guermat, and Tucker (2018) who rely on a GARCH
process to estimate several betas for a single asset, they assume that this parameter
takes on two or three values that are switche by the very Markov chain that drives
changes in the market volatility. They further assume that the news idiosyncratic to
each asset (i.e., the residuals of the CAPM) also follows a two-state Markov-switching
process that is independent of that of the market. To estimate their models by the
maximum likelihood method, they form portfolios using the returns of all stocks listed
on the NYSE, the AMEX, and the NASDAQ over the sample period ranging from
July 1963 to December 2010. They find supporting evidence for the Markov-switching
conditional CAPM and, performing diagnostic checks on the residuals of their models,
they also conclude that they capture the ARCH effects and the non-normalities oberved
in the data.
I use a slightly different and simpler approach to estimate in one go the CAPM for
ten of the sectors of the TSX. I assume that the market excess returns on the TSX
and the excess returns of its ten sectors follow a multivariate normal distribution whose
means and variance-covariance matrix depend on a single Markov process. This is a gen-
eralization of the model used by Vendrame, Guermat, and Tucker (2018). But, relying
on the evidence from the diagnostic checks on residuals performed by Abdymomunov
and Morley (2011), I assume that the volatility of each asset is constant within each of
the states of the market. Earlier attempts to estimate the CAPM from state-dependent
multivariate normal distributions include Tu (2010).
4.1 The Method of Estimation
Let’s consider the column vector Yt = [Rmt − Rf t , (Rst − Rf t 1)′ ]′ that lists the market
excess return and the distribution of the excess returns across the sectors of the TSX,
at time t. The joint probability of observing Yt is described by the state-dependent12 4 THE CONDITIONAL CAPM
multivariate normal distribution
− 11 1 − 12
′ −1
p Yt ; µSt , ΣSt = (2π) (detΣSt ) exp − Yt − µSt ΣSt Yt − µSt
2
2
(4.6)
µm,St Σm,m,St Σm,s,St
with µSt = and ΣSt = ,
µs,St Σm,s,St Σs,s,St
where the operator det denotes the determinant and the variable St in subscript denotes
either the first-order or the second-order Markov chain described by relation (3.1) or
(3.5).
The conditional systematic risks, β St , and the measure of performance, αSt , of the
sectors can be estimated simultaneously using the parameters of the state-dependent
multivariate normal distribution in (4.6).
β St = Σm,s,St Σ−1
m,m,St
(4.7)
αSt = µs,St − Σm,s,St Σ−1
m,m,St µm,St
The parameters µSt and ΣSt (St = 1, 2, . . . ) of the state-dependent multivariate
normal distributions are themselves estimated maximizing the likelihood of observing a
sample yt (t = 1, 2, . . . , T ) of the realized excess returns. The likelihood of the first-order
and the second-order Markov-switching models are
T X
Y 2
L (θ) = γst−1 ,st p (yt |st ; θ st )
t=1 st =1
T
Y
=u ΓP (yt ) 1′
t=1
with θ = γ11 , γ12 , γ21 , γ22 , µ′1 , µ′2 , vech(Σ1 )′ , vech(Σ2 )′ (4.8a)
T X
Y 2
L (θ ∗ ) = γs∗∗t−1 ,s∗t p yt |s∗t ; θ ∗st
t=1 st =1
T
Y
= u∗ Γ∗ P (yt ) 1′
t=1
∗
∗ ∗ ∗
with θ = γ11 , γ12 , . . . , γ44 , µ∗′ ∗′ ∗ ′ ∗
1 , . . . , µ4 , vech(Σ1 ) , . . . , vech(Σ4 ) , (4.8b)
where the variable s∗t , as explained in Section 3, is a transformation of a second-order
Markov chain into a first-order chain, the vector u, the stationary distribution of st ,
is defined by relation (3.4), Γ is the matrix of transition probabilities, and P (yt ) is
a diagonal matrix whose j-th diagonal element corresponds to either p (yt |st = j; θ)
or p (yt |s∗t = j; θ ∗ ). The operator vech denotes the half-vectorization of the variance-
covariance matrix (i.e. the transformation of this symmetric matrix into a column vector
by stacking only its lower trinagular elements)4.2 The Findings 13
I have estimated the parameters θ and θ ∗ by maximizing directly the natural log-
arithm of the likelihood (log-likelihood, in short) of the Markov-switching models. To
do that, I have used the base function nlm, which stands for non-linear minimization,
of the software R (www.r-project.org). The models have multiple maxima. Therefore,
for each of the two models, I have performed the numerical optimization hundred times,
with different starting parameter values generated randomly, in order to select the es-
timates that give the highest log-likelihood. Since repeating this several times gives
roughly the same estimates, I have concluded they are global maxima.
The estimated parameters are assumed to be asymptotically normal. Then, one
needs their standard errors to perform the tests of significance. A way of getting them
is by inverting the Hessian matrix (i.e., the matrix of the second derivatives of the
log-likelihood with respect to the parameters). 3 The Hessian matrix is computed
numerically at the maximum. Since some of the estimated parameters at the maximum
can lie on or close to the boundary of the parameter space, inverting the Hessain matrix
does not always yield finite numbers. To overcome this issue, I have performed some
bootstrapping using the estimated parameters to sample the explained variables rst −rf t 1
(t = 1, . . . , T ) one thousand times from state-dependent normal distributions. Then, I
have estimated new parameters using the simulated time series and the observed rmt .
The standard errors of the parameters are then computed as the standard deviation of
the bootstrap estimates. To perform the significance tests, I have computed z-statistics
by dividing the global maxima by the bootstrap standard errors.
4.2 The Findings
I present some evidence from estimating the expected excess returns and their variance-
covariance matrices using, in turn, the first-order Markov chain (i.e., maximizing the
likelihood given by relation (7.2a)) and the second-order Markov chain (i.e., maximizing
the likelihood given by relation (7.2b)). Note that both types of Markov chains have
two states. The resulting estimates of the conditional CAPM are also presented. The
data used to estimate the models are described in Appendix A.
4.2.1 The First-Order Markov-Switching CAPM
Figure 4.1 compares the kernel density curves of the excess returns across the TSX fo the
mixture of the state-dependent multivariate normal distributions fitted to these data.
The marginal distributions that are produced using the component expected values and
standard deviations of the Markov-switching model are close to the actual distributions
of the excess returns. However, the actual distributions in the financial, industrial, and
information technology sectors are somewhat taller than the ones fitted to the data.
Relation (4.9) shows the estimates of the transition probabilities of the two states of
the Markov chain and their stationary distribution. State 1 is the most likely one. It
occurs 64% of the time and state 2 occurs 36% of the time. As it appears in Figure 4.2,
n h 2 io−1
ln L(θ)
3
var(θ) = −E ∂ ∂θ∂θ ′ .4 THE CONDITIONAL CAPM
1 Consumer Discretionary 2 Consumer Staples 3 Energy 4 Financial 5 Gold
0.12
0.04
0.12
0.12
0.06
0.10
0.10
0.10
0.05
0.03
0.08
0.08
0.08
0.04
Probability
Probability
Probability
Probability
Probability
0.06
0.02
0.06
0.06
0.03
0.04
0.04
0.04
0.02
0.01
0.02
0.02
0.02
0.01
0.00
0.00
0.00
0.00
0.00
−20 −15 −10 −5 0 5 10 −15 −10 −5 0 5 10 −30 −20 −10 0 10 20 −30 −20 −10 0 10 −40 −20 0 20 40
Excess Return Excess Return Excess Return Excess Return Excess Return
6 Industrial 7 Information Technology 8 Materials 9 Telecommunication Serv. 10 Utilities
0.10
0.10
0.06
0.10
0.05
0.05
0.08
0.08
0.08
0.04
0.04
0.06
0.06
Probability
Probability
Probability
Probability
Probability
0.06
0.03
0.03
0.04
0.04
0.04
0.02
0.02
0.02
0.02
0.02
0.01
0.01
0.00
0.00
0.00
0.00
0.00
−20 −10 0 10 20 −40 −20 0 20 −40 −30 −20 −10 0 10 20 −20 −10 0 10 20 −20 −15 −10 −5 0 5 10
Excess Return Excess Return Excess Return Excess Return Excess Return
Low Volatility State High Volatility State Marginal Distribution
Figure 4.1: Kernel Density Curves of Returns across the TSX and their Stationary Markov-Dependent Mixture of Normal
Distributions.
144.2 The Findings 15
Expected Value Volatility
1.5
14
12.8
1.1
1.04
12
1.0
0.93 0.92
0.9
0.81
0.73
0.7
10
0.6
0.53 9.2
9
0.47
0.5
8.8
0.42
8
7.3 7.5
6.9
0.03
6.2 6.2
0.0
5.7
6
5.3 5.2
−0.24 −0.25 4.8
−0.34 4.5
−0.38
3.8
4
−0.5
−0.55
3.1 3.1 3.1
−0.65 2.8 2.9
2.7
−0.74
2
−0.88
−1.0
−1.18
0
CD CS E F G I IT M TS U Market CD CS E F G I IT M TS U Market
State 1 State 2
Figure 4.2: Expected Value and Volatility from a First-Order Markov-Dependent
Mixture of Multivariate Normal Distributions Fitted to Excess Returns across the
TSX, 1998:M1-2017:M12.
all the excess returns are less volatile in state 1 than in state 2. For example, the
volatility (or standard deviation) of the market excess return is 2.69 in state 1 and 6.11
in state 2. On the other hand, the expected excess returns across the TSX are higher in
state 1 than in state 2, except for the gold sub-industry. For example, in the market, the
monthly expected excess returns are .73% in state 1 and -.55% in state 2 while, in the
gold sub-industry, they are respectively -.34% and .7% in states 1 and 2. The expected
excess returns are positive in state 1 and negative in state 2 in the other sectors, except
in the consumer staples and the financial sectors where they are positive over the two
states.
.953 .047
(43.70) (2.17)
Γ̃ =
.084
.916
(2.14)2 (3.25) (4.9)
.640 .360
ũ =
(5.54) (3.12)
State 1, the low volatility state, can be labeled as the bull market. The reason is
that, in this state, the excepted excess returns are positive across the TSX, except for
the gold sub-industry. Consequently, state 2, the high volatility state, can be labeled
as the bear market. Decoding the states (i.e., for each time period, identifying using
the fitted model which of the two states is the most likely) reveals that the bear market16 4 THE CONDITIONAL CAPM
include mainly the period ranging from November 2007 to June 2009, which indeed
corresponds to the global financial crisis, and the early 2000s, which corresponds to the
dot-com crash.
Excess returns are not necessarily positive during bull markets, evenn though their
expected values are positive. Between 1998 and 2017, out of 156 months where the TSX
is very likely to experience a bull market trend, the market excess returns are positive 101
times. In the gold sub-industry and the materials sector, positive excess returns occur
respectively 76 and 82 times. The financial sector followed by the consumer discretionary
are the ones that show more positive excess returns during bull markets, respectively
106 and 105 times. Conversely, positive excess returns occur during bear markets, where
the market excess returns turn out to be negative only 40 times out of 83 months.
The estimates of the transition probability matrix reported in (4.9) indicate that
both state 1 and state 2 are very persistent. The probability of leaving state 1 is only
4.7%. Therefore, state 1, which is identified as the bull market, is the most recurrent
and the most persistent of the two states.
Note that all the estimates of the transition probabilities and the unconditional
probabilities of the states reported in (4.9) are statistically significant since their z-
values, which are displayed in parentheses, are greater than their 5% critical value,
which is 1.64.
In Table 4.1, I have reported the estimates of the parameters of the first-order
Markov-switching conditional CAPM. These parameters are the measure of performance
of Jensen (i.e., the intercept terms or the alphas) and the systematic risks (i.e., the slope
parameters or the betas). They are computed out of the component expected values and
variance-covariance matrices, using relation (4.7). The adjusted coefficients of determi-
nation, R̄2 , reported in the last column of this table have been computed after decoding
the consecutive states that most likely prevailed on the TSX over the sample period.
All the betas are significantly positive, except those of the gold sub-industry and the
utilities sector, in state 2. This means that, according to the first-order Markov-switching
model, in the bear market, the overall excess return on the TSX does not significantly
explain the excess returns in the gold sub-industry and in the utilities sector. The betas
of the financial sector are almost the same, over the two states, .766 in the bull market
and .75 in the bear market.
In bull markets, gold and more generally materials along with energy are high-risk
assets. Their betas are greater than the market beta, viz these estimates, which are
roughly equal to 1.5, are greater than 1. For this reason, they should offer the possibil-
ity of higher returns, but they deliver negative alphas, which lowers their theoretically
appropriate excess returns. In bull markets, the sectors that tend to outperform the
market are those having high betas and delivering, at the same time, positive alphas.
These sectors are the consumer discretionary, the financial, the industrial, the informa-
tion technology, and the telecommunication service sectors.
In the bear market, information technology is the only sector that has a beta greater
than 1. Furthermore, its alpha is negative. Thus, investing in this sector is unattractive,
since the market excess returns and, consequently, the theoretically required excess
returns in this sector are very likely to be negative, in bear markets. In bear markets,Vous pouvez aussi lire
